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Travelling wave solutions to the sine-Gordon equation for which the quantity c2 1 < 0 are called subluminal waves. Se hela listan på wiki2.org We obtain exact solutions U(x, y, z, t) of the three-dimensional sine-Gordon equation in a form that Lamb previously proposed for integrating the two-dimensional sine-Gordon equation. The three-dimensional solutions depend on arbitrary functions F(α) and ϕ(α,β), whose arguments are some functions α(x, y, z, t) and β(x, y, z, t). The ansatzes must satisfy certain equations. These are an Sine-Gordon Expansion Method for Exact Solutions to Conformable Time Fractional Equations in RLW-Class Alper Korkmaza;, Ozlem Ersoy Hepsonb, Kamyar Hosseinic, Hadi Rezazadehd, Mostafa Eslamie Approximate solutions for solving the Klein–Gordon and sine-Gordon equations Majeed A. Yousifa,*, Bewar A. Mahmoodb aDepartment of Mathematics, Faculty of Science, University of Zakho, Duhok, Kurdistan Region, Iraq bDepartment of Mathematics, Faculty of Science, University of Duhok, Duhok, Kurdistan Region, Iraq In this scheme we advance the solution of the (2+1) sine-Gordon partial differential Equation (1) from nth plane to (n+1)th plane by replacing u xx by implicit finite difference approximation at the (n+1)th plane. Similarly, u tt, u yy and u t are replaced by an explicit central finite difference approximation at the nth plane as in equations Klein–Gordon equation4,5 in some special cases. Many researchers focused on the solitary wave solutions of the sine-Gordon equation; here for the ﬁrst time, we will point out its periodic solution.
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The resulting Kosterlitz–Thouless phase diagram is One can obtain a $\phi^4$ model as an approximation of the sine-Gordon model, expanding the sine term in the equation or cosine term in the Hamiltonian. However, there is no general relation between the $\phi^4$ model and the SG model. In other words, one cannot map any solution of the one equation to that of the other. The Sine Gordon Equation In:=Clear@"Global One Soliton Solutions We display a soliton solution; first of all for the special case of no time depen- of the sine-Gordon equation when the underlying wave is a travelling wave. This is related to the work done in [DDvGV03], where stability of a singularly perturbed subluminal kink wave solution was shown. Travelling wave solutions to the sine-Gordon equation for which the quantity c2 1 < 0 are called subluminal waves. Ding, D.-J., et al.: Analytical Solutions of Differential-Difference Sine-Gordon Equation 1702 THERMAL SCIENCE, Year 2017, Vol. 21, No. 4, pp.
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2.1 Sine–Gordon equation (SGE). 12 Dec 2012 Title:Numerical Solutions to the Sine-Gordon Equation Abstract: The sine- Gordon equation is a nonlinear partial differential equation. It is known Exact, particular solutions of the double sine Gordon equation in n dimen- sional space are constructed. Under certain restrictions these solutions are N solitons and the field equations.
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Approximate solution of the fuzzy fractional Bagley-Torvik equation by the RBF Numerical solution of nonlinear sine-Gordon equation with local RBF-based Keyword: Optical solitons, Birefingent, Kerr law, sine-Gordon expansion method present in Section 2, Section 3, we find exact families of soliton pair solutions Glasgow Mathematical Journal, vol. 51A, ss.
The equa-tion, as well as several solution techniques, were known in the nineteenth century in the course of study of various problems of differential geometry. The equation
The sine-Gordon equation which includes the d’Alembert operator and the sine of the unknown function is used in explaining a number of phenomena including the propagation of fluxons in Josephson junctions, dislocations in crystals and the motion of rigid pendula attached to a …
pseudospherical surfaces are solutions of the sine-Gordon equation, the B acklund transform can be viewed as transforming solutions to the sine-Gordon equation. In particular, if u is a solution of the sine-Gordon equation, u xy= sinu, then the auto-B acklund transformation is the system v x= u x+ 2asin(u+ v 2) v y= u y+ 2 a sin(v u 2)
The Sine Gordon Equation In:=Clear@"Global‘*"D The Equation The equation is ••••••••• ¶2u ¶x2-••••••••• ¶2u ¶t2 =sinu In:=sinegordoneq@u_D:=D@u@x,tD,8x,2
The magnetic flux trapped in a long Josephson function (two bulk superconductors seperated by a narrow layer of a dielectric material) solution of sine-Gordon equation in circular domain. The global kernel method resulted the dense differentiation matrices and hence difficult to apply for problem with large amount of data points. The present numerical scheme is local with sparse differentiation matrices, consequently capable of removing the deficiency of ill- Based on a new discrete Lax pair, an elementary explicit solution is found for the lattice sine-Gordon equation through Riemann surface method.
The Sine-Gordon expansion method is implemented to construct exact solutions some conformable time fractional equations in Regularized Long Wave (RLW)-class. Compatible wave transform reduces the governing equation to classical ordinary differential equation. The rest of this paper is rearranged as follows: In Section 2, the details of the sine-Gordon expansion method are presented. In Section 3, the sine-Gordon expansion method is implemented to produce the exact traveling wave solutions of the Tzitzeica type equations in nonlinear optics; and finally, a short conclusion is provided in Section 4.
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Soliton is a kink which changes the Josephson phase from 0 to 2π (soliton) or from 2π to 0 (anti-soliton). The ﬁeld of soliton is h = φ x = 2 cosh(√x−ut 1−u2),h| x=0 =2 For other exact solutions of the sine-Gordon equation, see the nonlinear Klein–Gordon equation with f(w) =bsin(‚w). 5–. The sine-Gordon equation is integrated by the inverse scattering method. References Steuerwald, R., Uber enneper’sche Fl¨ achen und B¨ ¨acklund’sche Transformation, Abh. Bayer. Akad.
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Based on nonlinear spectral analysis, exact nonreflecting boundary conditions are derived at two artificially introduced boundary points. Then a numerical scheme of second order is proposed to approximate the solution. D. Kaya, An application of the modified decomposition method for two dimensional sine-Gordon equation, Applied Mathematics and Computation, 159(2004): 1-9.
Gordon McGlathery is a Training Solutions Consultant at Symetri based in our Manchester office. His day-to-day work is all about making people and technology Creating an ideal solution for evoking a feeling of being near/in water. Case studies // Sunlight into architecture Inspired by Gordon Matta-Clark placing sand on top of the plastic membrane and playing sine wave tones of The book includes exercises and solutions.